A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research

Transcription

1 Journal of Asia Pacific Studies (2018) Volume 5 Issue 1, 1-61 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research *Dr. Meena Madhavan, Lecturer-Faculty of Business Administration, St.Theresa International College, Thailand meena2priya@gmail.com Abstract The purpose of this study is to facilitate beginners to understand mediation analysis and its statistical procedures in the field of management. This article explained the evolution of mediation analysis, which provides a basic understanding towards the idea of mediation. And further, the general assumptions to carry out mediation analysis were highlighted. Next, the study covered the different approaches towards mediation; and statistical procedures were outlined to ease the understanding of practicing researchers in the field of management. And it was noted that, bootstrapping is the best method to conduct mediation analysis. One of the major limitations of this study is, the technical part of the statistical theory about variance, estimates, confidence intervals, effect size, etc., were not explained in detail. Also, few modern methods were not included as the motive is to ease the understanding of practicing researchers in the field. Keywords: mediation analysis; traditional method; modern methods; hypothesis testing 1

2 Dr. Meena Madhavan Introduction The purpose of this study is to understand the mediation analysis and statistical tests used to investigate the mediation effects. Scholars in the field of Psychology and Social science widely use mediation analysis to test the causality. There is a high demand among researchers to test the causal effects of the intermediate variable, which exert influence on dependent variable. The mediation analysis is often conducted to examine the type of relationship or effects between independent variable and dependent variable by using the proposed mediating variable. The causal mechanisms assess the indirect effects produced by predictor variable. This causal chain is referred as mediation analysis. Baron and Kenny (1986) developed the causal chain model to test the mediation effects. This model is widely used in social science research. Baron and Kenny s model was popularized in the field of social science research with citations (according to Google Scholar). Zhao, Lynch and Chen (2010) pointed out that many research projects was revoked at the early stages or staggered at the finishing stage as it was not conformed to Baron and Kenny s condition. The authors presented the nontechnical flaws in the Baron and Kenny s logic and also provided the alternative decision-tree & step-by-step framework for mediation tests. Many researchers have criticized Baron and Kenny s approach with valid logic; but still it is popular in the field. One of the possible reasons may be that the practicing researchers develop their basic understanding to mediation tests by following the traditional Baron and Kenny s approach. Nevertheless, there are modern technological advancements with advanced statistical tools dismiss the use of older mediation tests. The researchers are curious to apply the statistical advancements and consider it as meaningless to use the obsolete tests. This study would ponder out the different approaches i.e. from traditional Baron and Kenny to modern SEM model to ease the understanding of practicing researchers/scholars in the field of Management, which will also cover the application of mediation tests and its statistical procedures. 2

3 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research Mediation Analysis This part aims to answer the following questions: 1. What is mediation analysis? 2. Why mediation analysis? 3. What is the difference between mediating and moderating variables? 1. What is mediation analysis? Mediation analysis is more prominent in psychological research (Mackinnon, Fairchild, and Fritz, 2007). The use of mediation analysis in the field of psychology prompted from stimulus-response formula, in which it is stated as "Mediating refers to the possibility that the process may act as a link between a sensory input and a response not directly connected with itone main function of such a process. Also in other words mediating processes are fundamentally a means of modifying the way in which sensory control acts, not an absence of it. This was supported with various examples and experiments in the text book of Psychology by Hebb (1958). Hence, the mediation process or analysis rooted from the field of psychology. Kenny (2018 & n.d.) stated that, the history of mediation tests emerged from the researchers Wright (1934), Fisher (1935), and Hyman (1955). In general mediation analysis is referred as the mechanism to study the cause and effect relationship between the predictor variable (X) and dependent variable (Y), where a mediating variable is hypothesized to intermediate the relationship between X and Y. In other words, the effect of X on Y is intervened or mediated by the mediating variable M and still the causal variable X affects the dependent variable Y (Kenny, 2018). The most common approaches widely used by researchers for testing the meditational hypothesis are Sobel test (Sobel, 1982) and Baron & Kenny s 4 step process (Baron and Kenny, 1986). Later, these tests were considered as obsolete by other researchers in the field due to the raise of modern approaches. And few researchers have compared the models and reported that traditional mediation approaches have low statistical power when compared to the modern approaches (MacKinnon, Lockwood, Hoffman, West, & Sheets, 3

4 Dr. Meena Madhavan 2002; Biesanz, Falk, & Savalei, 2010). The general understanding to mediation process is depicted in the following figure (Kenny, 2018): Figure 1 From the Figure 1 it can be understood that X is the causal variable and that causes Y the outcome. This is unmediated model and path c in the above model is called the total effect (Kenny 2018). Source: Kenny (2018) Figure 2 is the mediated model, the effect of X on Y is mediated by a mediating or intervening variable M and still X may cause Y. The path c in the figure 2 is called the direct effect. To probe the evidence of mediation it is important to demonstrate that the effect of the treatment on the outcome variable is zero after the mediator is controlled (Judd and Kenny, 1981). Complete mediation would happen when X no longer affects Y after 4

5 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research mediating variable M has been controlled and making path c zero. Partial mediation would occur when the mediator is introduced the effect of X to Y is reduced or different from zero. Mediation model is the causal model and it is presumed to cause the outcome Y (Kenny, 2018). When Mediation Analysis should be used? Major assumptions for mediation analysis as follows: 1. The researchers research scope is about testing the relationship between three variables, among which one is intervening or mediating variable. Judd and Kenny (1981) stated that it is necessary to follow the process analysis to specify the causal chain that is responsible for treatment effects. This analysis has value in evaluation research for three reasons: firstly, it examines and specifies the causal mechanisms that produce outcomes; secondly, once the theoretical model has been framed for the outcome behavior it is easy to generalize the results in other research settings; thirdly, the researcher knows the process and variables that have direct impact. Also the authors mentioned, in order to claim the mediation there must be three conclusions: 1. the predictor variable causes outcome variable, without this there is no mediation, 2. the predictor variable causes potential mediator, and 3. the mediator must cause the outcome variable controlling for the predictor variable, unless it directly affects the outcome variable it can t be claimed as mediator. It is considered to have mediation effects if there is evidence for the above three conclusions. 2. The mediating hypothesis should be framed accordingly with the proper theoretical support stating the relationships with X or Y or based on its practical applicability. If there is no theoretical support, the final results would be opposite as no mediation and this will not meet the stated research scope/objectives/hypothesis. 5

6 Dr. Meena Madhavan 3. The researchers should not confuse the mediation and moderation analysis. The mediation analysis is the analysis which explains the relationship between X and Y, whereas the moderation analysis influences the relationship between X and Y. Baron and Kenny (1986) stated that a moderator is a qualitative (e.g., sex, race, class) or quantitative (e.g., level of reward) variable that affects the direction and/or strength of the relation between an independent or predictor variable and a dependent or criterion variable. And also a moderator variable is introduced when there is a weak or inconsistent relationship between predictor and criterion variable, whereas mediator is introduced when there is a strong relation between the predictor and criterion variable. The understanding towards mediating variable is given in figure Judd and Kenny (2010) stated that it is necessary to have the valid causal assumptions for the mediation to be valid. 5. It is important to consider the standard assumptions for the general linear model such as linearity, normality, homogeneity of error variance, and independence of errors (Kenny, 2018). Full Mediation and Partial Mediation The term full mediation is also referred as complete mediation. Baron and Kenny (1986) stated this as perfect mediation, the situation where the independent variable has no effect when the mediator is controlled. Hair, Black, Babin, Anderson, and Tatham (2006) stated that full mediation is where the relationship between predictor variable and outcome variable becomes insignificant after the inclusion of mediating variable. Partial mediation is where the effect of relationship between predictor and outcome variable is reduced and still it is significant after inclusion of the mediating variable. Kenny (2018) stated the difference between complete and partial mediation as follows: Complete mediation is the case in which variable X no longer affects Y after M has been controlled, making path c' zero. Partial mediation is the case in which the path from X to Y is reduced in absolute size but is still different from zero when the mediator is introduced. 6

7 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research Statistical analysis for testing mediation effects Regression It was suggested by the researchers Judd and Kenny (1981), James and Brett (1984), Baron and Kenny (1986) to conduct the series of regression equations to identify the mediation effects with four step approaches. The equations could be framed accordingly (Testing Mediation with Regression Analysis, 2017). 7

8 Dr. Meena Madhavan Difference in these approaches has been discussed by Kenny (2018) that James and Brett (1984) specified that it should not be controlled by the X (causal variable) and assumed the full mediation implicitly, whereas Judd and Kenny (1981) & Baron and Kenny (1986) control for X in step 3. In Baron and Kenny s approach step 4 is not required and Judd and Kenny s approach include all four steps (Kenny, n.d). For the first 3 steps simple linear regression should be conducted and for the step 4 multiple regression should be conducted (Testing Mediation with Regression Analysis, 2017). If all the four steps are met, it indicates M completely mediates the relationship of X and Y and the first three steps are met and step 4 is not met then it indicates partial mediation (Kenny, 2018). Example: If the researcher would like to test the hypothesis using below concept model about the supply chain management practices of the manufacturing industry and its impact on the business performance. Here, the mediating variable is introduced i.e. competitive advantage, because in the recent times the link between supply chain management and strategic management has been addressed by many studies. Testing the model for reliability or validity is not conducted here, as the research area is already existing in pace and the main objective of the study is to discuss about the statistical procedures of mediation analysis. Here X is Supply Chain Management (Predictor/Independent Variable), Y is Business Performance (Criterion/Dependent Variable), and M is Competitive Advantage (Mediating Variable). In this example, there are 4 observed variables for X (Supply Chain Management), 5 observed variables for M (Competitive Advantage), and 4 observed variables for Y (Business Performance). 8

9 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research Let s assume that the sample respondents for the study were around 286, which is sufficient to run the advanced statistical models. The dummy dataset were prepared for the study. Also, the following hypotheses were constructed appropriately for the study. Hypothesis 1: Supply chain management practices of the firms impact on business performance. Hypothesis 2: Supply chain management practices of the firms impact on competitive advantage. Hypothesis 3: Competitive advantage of the firms impact on business performance. The following are the procedures based on Baron and Kenny; the simple linear regression was performed using SPSS 25 Trial version. The results are presented below for all three hypotheses: For the Hypothesis 1: Supply chain management practices of the firms impact on business performance. The effect of X (Supply Chain Management) on Y (Business Performance) is assessed in the step 1. If the results are not significant, there may be no possibilities for mediation. After calculating the mean for all 4 observed variables in Supply Chain Management (X) and 4 observed variables in Business Performance (Y), the 9

10 Dr. Meena Madhavan variable Supply Chain Management (X) and Business Performance (Y) has been entered in the appropriate boxes below. Now, after clicking the Ok button the following results were displayed in the SPSS output window. Table 1.1 Model Summary Adjusted R Std. Error of Model R R Square Square the Estimate a a. Predictors: (Constant), SCM 10

11 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research Table 1.2 ANOVA a Sum of Model Squares df Mean Square F Sig. 1 Regression b Residual Total a. Dependent Variable: BP b. Predictors: (Constant), SCM Table 1.3 Coefficients a Unstandardized Coefficients Standardized Coefficients Model B Std. Error Beta t Sig. 1 (Constant) SCM a. Dependent Variable: BP For the Hypothesis 2: Supply chain management practices of the firms impact on competitive advantage. In this step competitive advantage M is regressed against supply chain management X. So while writing it we state competitive advantage as Y, because it is the dependent variable here. The effect of X (Supply Chain Management) on Y (Competitive Advantage) is assessed in the step 2. After calculating the mean for all 4 observed variables in Supply Chain Management (X) and 5 observed variables in Competitive Advantage (M), the variable Supply Chain Management (X) and Competitive Advantage (M) has been entered in the appropriate boxes below. 11

12 Dr. Meena Madhavan Now, after clicking the Ok button following results were displayed in the SPSS output window. Table 1.4 Model Summary Adjusted R Std. Error of Model R R Square Square the Estimate a a. Predictors: (Constant), SCM Table 1.5 ANOVA a Sum of Squares Df Mean Square F Sig. Model 1 Regression b Residual Total

13 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research a. Dependent Variable: CA b. Predictors: (Constant), SCM Table 1.6 Coefficients a Unstandardized Coefficients Standardized Coefficients Model B Std. Error Beta t Sig. 1 (Constant) SCM a. Dependent Variable: CA For the Hypothesis 3: Competitive advantage of the firms impact on business performance. In this step business performance Y is regressed against competitive advantage M. So while writing it we state competitive advantage as X, because it is the independent variable here. The effect of X (Competitive Advantage) on Y (business performance) is assessed in the step 3. In the previous two hypotheses the mean for all 5 observed variables in Competitive Advantage (M) and 4 observed variables in Business Performance (Y) has been calculated. So, now just the researcher should enter the variables in appropriate boxes as shown below. 13

14 Dr. Meena Madhavan Now, after clicking the Ok button following results were displayed in the SPSS output window. This is the simple regression and it is not required to click any other options like statistics, options, etc. The researchers can go with the default selection in the statistics tab in the right hand side (in first above the plots option). Table 1.7 Model Summary Adjusted R Std. Error of Model R R Square Square the Estimate a a. Predictors: (Constant), CA 14

15 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research Table 1.8 ANOVA a Sum of Model Squares Df Mean Square F Sig. 1 Regression b Residual Total a. Dependent Variable: BP b. Predictors: (Constant), CA Table 1.9 Coefficients a Unstandardized Coefficients Standardized Coefficients Model B Std. Error Beta t Sig. 1 (Constant) CA a. Dependent Variable: BP The below is the summarized table (Step 1, Step 2, and Step 3) of all three simple linear regressions for the stated hypotheses. This format may be followed when the researchers are reporting it into the dissertation. Summarized Table

16 Dr. Meena Madhavan Hypothesis Predictor/ Independe nt Variable X Dependent Variable Y a intercept value B Standard error b β t-value p-value Hypothesis Support Hypothe sis 1 Hypothe sis 2 Hypothe sis 3 Supply Chain Manageme nt a Supply Chain Manageme nt b Competiti ve Advantage c Business Performa nce Competiti ve advantage Business Performa nce Yes * Yes * Yes * a R 2 =.045, b R 2 =.063, c R 2 =.223, P 0.01*, n = 286 From the above simple linear regression and summary tables, it is observed that all the three stated hypotheses were supported. Using Baron and Kenny s approach is to find out at the initial stage whether the independent variable is the significant predictor of dependent variable. It is evident from the p-value, which is statistically significant at p<0.01. Following the three stage approach, at the first step business performance was regressed against supply chain management (Hypothesis 1); it is observed that supply chain management has significant and positive impact (b =.213, p<0.01) on business performance F (1,284) = , p<0.01. In the second step, competitive advantage was regressed against supply chain management (Hypothesis 2); it is observed that supply chain management has significant and positive impact (b =.252, p<0.01) on competitive advantage F (1,284) = , p< In the third step, business performance was regressed against competitive advantage (Hypothesis 3); it 16

17 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research is observed that competitive advantage has significant and positive impact (b =.472, p<0.01) on business performance F (1,284) = , p<0.01. And also the R square values indicate that the independent variable supply chain management accounted for 4% of the variance in business performance and supply chain management accounted for 6% of the variance in competitive advantage, whereas the variable competitive advantage accounted for 22% of the variance in business performance. It is evident that the Mediator M is having considerable amount of impact when it is regressed against business performance compared to supply chain management. At this stage, the researcher can assess whether the mediator is statistically significant to run the advanced model, although it was claimed by many researchers that Baron and Kenny s approach is obsolete. Hence, this approach gives clear idea to the researchers interested in doing mediation analysis. Multiple Regression Using multiple regression analysis researchers are interested in identifying the best predictors. And also there is a need to identify those predictors that are supportive of theory. The two approaches which could deduct the best predictors are stepwise regression and hierarchical regression. Hierarchical regression would analyze the effect of predictor variables after controlling for other variables (Lewis, 2007). The researchers can choose the entry of variables in each step based on the theory, the variable which they would like to control. In the first step, the variable which is controlled could be entered. And in the second step, other variables could be entered. Depending on the experiment design of the researcher, the variables could be either entered separately in the each step or based on the hierarchy it could be entered with supporting theory. Also, the researchers may opt for stepwise methods if they are interested in identifying the predictors that are most effective instead of enter method, although it depends on the researchers experiment design. Wampold and Freund (1987) stated that hierarchical regression is specifically used to test the theory based hypothesis. 17

18 Dr. Meena Madhavan Continuing with the Example and Step 4, Hierarchical multiple regression is conducted. The procedures of SPSS are given below: The major purpose for performing hierarchical regression in mediation test is to assess is there any multicollinearity effects and to analyze the effect of predictor variable after controlling for M. Researchers also use hierarchical regression basing Baron and Kenny s approach for testing mediation by analyzing the amount of variance in R 2 after introducing the mediating variables. But here we assess the multicollinearity effects between the two variables (predictor and mediator) in determining the level of dependent variable. Multicollinearity Effects Multicollinearity refers to the situation where two or more explanatory variables are highly linearly related (Hawking, 1983). Perfect multicollinearity is where the relationship between two independent variables is equal to +1 or -1. This occurs in rare datasets due to the redundancy of information. Once the items with redundancies are removed it will be free from multicollinearity effects. In other words, it can be stated as the correlation between two or more explanatory variables is larger than the correlation between the predictor and criterion variables, the perfect multicollinearity exists (Klein, 1962). Basically, multicollinearity problem can be detected under following cases: a. large change in the regression coefficients when the new variables is added or removed, b. insignificant 18

19 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research regression coefficients for a variable and rejecting the joint hypothesis those coefficients are statistically significant, c. Hair et. al (2006) mentioned that when pearson correlation coefficient between two independent variables are above 0.8, d. O Brien (2007) & Hair et al (2006) mentioned about the threshold levels of value of tolerance and variance inflation factor, which value of tolerance less than 0.20 or 0.10, and whereas variance inflation factor above 5.00 or indicates a serious problem of multicollinearity. And also the condition index value above 30 indicates the problem of multicollinearity. 19

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23 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research Select the above items in statistics click continue and then click ok. The results of multiple hierarchical regression are displayed below. Table 2.1 Model Summary Std. Change Statistics Adjusted Error of R R R the Square F ModelR Square Square EstimateChange Change df1 df a b a. Predictors: (Constant), CA b. Predictors: (Constant), CA, SCM Sig. F Change 23

24 Table 2.2 ANOVA a Dr. Meena Madhavan Sum of Squares df Mean Square F Sig. Model 1 Regression b Residual Total Regression c Residual Total a. Dependent Variable: BP b. Predictors: (Constant), CA c. Predictors: (Constant), CA, SCM Table 2.3 Coefficients a Unstandardized Coefficients Standardized Coefficients Collinearity Statistics Std. Model B Error Beta t Sig. ToleranceVIF 1 (Constant) CA (Constant) CA SCM a. Dependent Variable: BP 24

25 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research Table 2.4 Excluded Variables a Collinearity Statistics Model Beta In t Sig. Partial Correlation ToleranceVIF 1 SCM.101 b a. Dependent Variable: BP b. Predictors in the Model: (Constant), CA Minimum Tolerance Table 2.5 Collinearity Diagnostics a Condition Variance Proportions Model Dimension Eigenvalue Index (Constant) CA SCM a. Dependent Variable: BP The below is the summarized table for multiple hierarchical regression, which the researchers normally use for dissertation writing. 25

26 Dr. Meena Madhavan Table No. 2.6 (Summarized Table) Hierarchical Regression results of Business Performance against Competitive Advantage and Supply Chain Management Variables B Std. Error β ToleranceVIF Step (Constant) Competitive Advantage ** Step (Constant) Competitive ** Advantage Supply Chain Management * R 2 =.223 for Step 1: Δ R 2 =.009 for Step 2. *P 0.05, **P 0.01, n=286. The following is the general APA write-up for hierarchical regression. The hierarchical regression was performed to predict the business performance. The regression results revealed that competitive advantage has significant and positive impact (b=.472, p<0.01) on business performance F (1, 284) = , p<0.01. The multiple correlation coefficient R was at.472 and R square value indicated that competitive advantage accounted for 22% variance in business performance in the first step. And in the second step it is observed that competitive advantage has significant and positive impact (b=.447, p<0.01) on business performance F (2, 283) = , p<0.01. The multiple correlation coefficient R was at.482 and R square value indicated that competitive advantage accounted for 23% of variance in business performance in the second step. (This is the write up from tables 2.1, 2.2 and 2.3; F value and degrees of freedom could be found in ANOVA table 2.2, b (beta) values could be found in coefficients table 2.3, R and R square values could be found in Model summary table 2.1.) 26

27 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research The multicollinearity effects can be analyzed using the collinearity statistics given in table 2.3. It shows the value of tolerance between competitive advantage and supply chain management is.937, which is 0.20, indicates that there is no problem of multicollinearity. And it is further substantiated with Variance of Inflation (VIF) value. The VIF value between competitive advantage and supply chain management is 1.068, which is 5.0 indicates that there is no problem of multicollinearity. Therefore, both the mediating and predictor variable, competitive advantage and supply chain management doesn t influence each other in determining the level of business performance. For General Understanding Table 2.4 shows the excluded variables from the model. Here the variable supply chain management is excluded in model 1 and competitive advantage alone entered. Table 2.5 shows the collinearity diagnostics, which indicates relationship between variables and how they vary each other. The values above 1 in condition index indicate the correlation between two or more predictor variables. The values greater than 15 indicate a problem, whereas values of 1 are independent (Stepwise Linear Regression, n.d.). Indirect Effects The amount of variation is called as indirect effect. It measures the indirect effect or ab. The total effect can be written as, Total Effect = Direct Effect + Indirect Effect; it can also be denoted in symbols as c = c + ab. It also equals the reduction of the effect of the causal variable on the outcome or ab = c - c (Kenny, 2018), which is indirect effect = total effect direct effect. Difference of Coefficients and Products of Coefficients Approach 27

28 Dr. Meena Madhavan Baron and Kenny approach should be supplemented by the difference of coefficients or products of coefficients approach; because most of the researchers fail to calculate the indirect effects. Also, Baron and Kenny s approach failed to analyze the true mediation effects, which causes type II errors (Mackinnon, Fairchild and Fritz, 2007). To estimate the indirect effects the difference of coefficients and products of coefficients method could be used (Testing mediation with regression analysis, n.d.). The Difference of coefficients approach is proposed by Judd and Kenny (1981). Judd and Kenny recommended finding the difference between of regression coefficients. Referring to the equation 1 and equation 4 in this paper, it is subtracting the partial regression coefficients B1 (equation 4) obtained through multiple regression from the coefficient B obtained from simple linear regression (equation). This can be written as B indirect = B B1 (Testing mediation with regression analysis, n.d.). The products of coefficients approach was proposed by Sobel (1982). It is the product of regression coefficients obtained from two regression models i.e. Equation 4 and Equation 2 in this paper. This could be written as B indirect = (B2)(B). This approach is about X and M relationship, which is different from the difference approach. The partial regression coefficient for M predicting Y is referred as B2, whereas the coefficient from simple linear regression X predicting M is referred as B. The products of two regression coefficients reveal the indirect effects (Testing mediation with regression analysis, n.d.). Both Judd and Kenny s difference of coefficients approach and Sobel s products of coefficients approach produce identical values (Mackinnon, Warsi, & Dwyer, 1995). The difference of coefficients approach is about X and Y relationship and the products of coefficients approach is about X and M relationship. Procedure 28

29 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research The unstandardized regression coefficients from the linear and multiple regression should be included for the analysis. Calculation from the example, refer to the linear and multiple regression in this paper. Sobel Test Sobel test was proposed by Sobel (1982), and is often referred by researchers as Delta Method. Sobel test is more conservative and has low power (Mackinnon, Warsi, & Dwyer, 1995). This is because the sampling distribution of ab is highly skewed Kenny (2018). Sobel test is computed from the regression coefficients and standard errors. In order to determine the statistical significance of the indirect effect, a statistic based on the indirect effect must be compared to its null sampling distribution. The Sobel test uses the magnitude of the indirect effect compared to its estimated standard error of measurement to derive a t statistic (Sobel, 1982). Alternatively z or t distributions could be used to determine the significance (Mackinnon, Lockwood, Hoffman, West, and Sheets, 2002). Procedure The researchers can refer to the following link for calculating Sobel test (Preacher and Leonardelli, 2001). The webpage serves as an interactive calculation tool for mediation tests developed by Kristopher J. Preacher and Geoffrey J. Leonardelli. 29

30 Dr. Meena Madhavan The formulae for all three versions of mediation test were given below by referring to the above link and Mackinnon, Warsi and Dwyer (1995). All the three versions are performed using the above link. The usual Sobel test (Sobel, 1982) omits the third denominator, next the test that one adds the third denominator was popularized by Baron and Kenny (1986) is Aroian Test (1944/1947), whereas the Goodman tests Goodman (1960) subtracts it, and all three tests were tested in this example. Sobel Test z value = ab/ (b 2 S a 2 + a 2 S b 2 )... Equation (5) Aroian Test z value = ab/ (b 2 S a 2 + a 2 S b 2 + S a 2 S b 2 ).... Equation (6) Goodman Test z value = ab/ (b 2 S a 2 + a 2 S b 2 S a 2 S b 2 ).. Equation (7) Mackinnon, Warsi and Dwyer (1995) stated that Sobel and Aroian tests perform well. Researchers may enter the values of a and b from respective regression coefficient table to measure the indirect effect or mediation is statistically significant or not. Considering the sample example in this paper and following the above given link, Sobel test and other tests has been calculated. 30

31 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research The results can be written as: Table 3.1 Input Test Test Statistic Standard Error P-Value a =.254 Sobel Test b =.475 Aroian Test Sa =.058 Goodman test Sb =.057 a = raw regression coefficient (unstandardized) for the association between independent variable and mediator 31

32 Dr. Meena Madhavan Sa = standard error of a b = raw regression coefficient (unstandardized) for the association between mediator and predictor variable, when independent variable is also a predictor of dependent variable. Sb = standard error of b Interpretation The raw regression coefficient for the supply chain management (SCM) and competitive advantage (CA) is.254 (a) with a standard error of.058 Sa. The raw regression coefficient for the competitive advantage (CA) and business performance (BP) is.475 (b) with a standard error of.057 Sb, when independent variable supply chain management is also a predictor of business performance. The test statistic for Sobel test is 3.876, Aroian test is 3.854, and Goodman test is with an associated p-value of The standard errors of Sobel, Aroian, and Goodman test are 0.031, mostly identical for all the three tests. The z test value is >1.96 with an associated p-value of <0.05 indicates that the relationship between supply chain management and business performance is mediated by competitive advantage. Hence, there is evidence of complete mediation or indirect effects. The indirect effects are statistically significant. Bootstrapping Method In order to further corroborate the mediation test, bootstrapping method could be considered to measure the direct and indirect effects effectively. Bootstrapping method by professor Preacher & Hayes (2004) performs well, when compared to Baron and Kenny s approach and Sobel s test; because Baron and Kenny s approach and Sobel s test has several criticisms regarding the sample size and type II error. Bootstrapping is a resampling method which is used to estimate confidence interval for indirect effects (Preacher and Hayes, 2004). It can be performed using SPSS Macros, Mplus, R package Lavaan, and Amos. The bootstrapping method can be simply used in SPSS by using process macro 2.16 version written by Hayes (2013). 32

33 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research Bootstrapping method is popular and used by many researchers for measuring indirect effects (Bollen and Stine, 1990; Shrout and Bolger, 2002). Bootstrapping method is a non-parametric method by resampling with replacement eg times. From each of these samples the indirect effect is computed and a sampling distribution can be empirically generated. Because the mean of the bootstrapped distribution will not exactly equal the indirect effect a correction for bias can be made. With the distribution, a confidence interval, a p value, or a standard error can be determined (Kenny, 2018). If power of the indirect effect is the major concern bias-corrected bootstrap should be used, and if type I error is the major concern percentile bootstrap method is suggested (Hayes and Scharkow, 2013). Basically, Model 4 could be used in Process macro by bootstrapping of 5000 samples in the given example. Model 4 allows up to 10 mediators operating in parallel. Process macro could be downloaded from the following link: Templates of conceptual model are available in the download folder of process macro. The models could be chosen according to the study objective. In this paper, the example has single mediator, hence model 4 has been chosen and demonstrated with steps. It is easy to install the PROCESS MACRO after download, click utilities under extensions tab, then select Install Custom Dialog (Compatibility mode). 33

34 Dr. Meena Madhavan After selecting the Install Custom Dialog (Compatibility mode), the following window will appear. 34

35 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research After installation, the macro could be found in SPSS under the following tab. 35

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38 Dr. Meena Madhavan Referring to the example in this paper, the variables should be entered into the respective boxes, SCM (Supply Chain Management) should be entered in the independent variable X box, BP (Business Performance) should be entered in outcome variable Y box, and CA (Competitive Advantage) in M variable box. Then, Model number, bootstrap samples and confident intervals are selected in default. So, based on the example the default selection is more appropriate to use. 38

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40 Dr. Meena Madhavan After clicking options, the box PROCESS Options will be appeared, then the respective boxes should be checked to measure the total effects, indirect effects, and direct effects (Shown below). 40

41 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research Matrix Run MATRIX procedure: ***** PROCESS Procedure for SPSS Release **** Written by Andrew F. Hayes, Ph.D. Documentation available in Hayes (2013). ************************************************************ Model = 4 Y = BP X = SCM M = CA 41

42 Dr. Meena Madhavan Sample size 286 ************************************************************ ************** Outcome: CA Model Summary R R-sq MSE F df1 df2 p Model coeff se t p LLCI ULCI constant SCM ************************************************************ Outcome: BP Model Summary R R-sq MSE F df1 df2 p Model coeff se t p LLCI ULCI constant CA SCM ******* TOTAL EFFECT MODEL ************* 42

43 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research Outcome: BP Model Summary R R-sq MSE F df1 df2 p Model coeff se t p LLCI ULCI constant SCM ******** TOTAL, DIRECT, AND INDIRECT EFFECTS ****** Total effect of X on Y Effect SE t p LLCI ULCI Direct effect of X on Y Effect SE t p LLCI ULCI Indirect effect of X on Y Effect Boot SE BootLLCI BootULCI CA Partially standardized indirect effect of X on Y Effect Boot SE BootLLCI BootULCI CA Completely standardized indirect effect of X on Y Effect Boot SE BootLLCI BootULCI 43

44 Dr. Meena Madhavan CA Ratio of indirect to total effect of X on Y Effect Boot SE BootLLCI BootULCI CA Ratio of indirect to direct effect of X on Y Effect Boot SE BootLLCI BootULCI CA R-squared mediation effect size (R-sq_med) Effect Boot SE BootLLCI BootULCI CA Normal theory tests for indirect effect Effect se Z p ******* ANALYSIS NOTES AND WARNINGS ****** Number of bootstrap samples for bias corrected bootstrap confidence intervals: 5000 Level of confidence for all confidence intervals in output: NOTE: Kappa-squared is disabled from output as of version END MATRIX The above displayed is the results from process macro by bootstrapping of 5000 samples. Now, the researcher should be able to interpret the above results about the indirect effects and direct effects. Interpretation 44

45 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research Considering the example discussed in this paper, the following interpretation has been made for the understanding about mediation effects. Table 4.1 Direct, Indirect and Total Effects Interactions Path Coefficient Standard P- t-value Error Value b(yx) c b(mx) a b(ymx) b b(yxm) c' The results are summarized in the above table according to their paths. The coefficient values, standard error, t-value, and p-value were taken from the matrix table respectively based on their paths. The variables were regressed accordingly and the results could be interpreted as the following. The mediation effects was measured using process macro at 95% confidence interval with 5000 bootstrap resamples (Preacher and Hayes, 2008). It is observed from the results that supply chain management (SCM) practices was positively associated with business performance (BP) (b=.2287, t (284) = , p<0.01). Also, it is observed that supply chain management (SCM) practices was positively associated with competitive advantage (CA) (b=.2542, t (284) = , p<0.01). It is revealed from the results that the mediator competitive advantage (CA) is positively associated with business performance (BP) (b=.4750, t (283) = , p<0.01). It is noted from the results that the paths a and b are statistically significant, hence the basic criteria has been satisfied. The bias-confidence interval estimates has been used as default (Preacher and Hayes, 2004; Mackinnon, Lockwood, and Williams, 2004). It is evident from the results that competitive advantage (CA) mediates the relationship between supply chain management (SCM) and business performance (BP) i.e. (b = c c is.1207, Standard Error =.0360, and Confidence Interval CI =.0576 to.2009 (CI should be different from zero)), which indicates that the indirect effects are statistically 45

46 Dr. Meena Madhavan significant. Although, the results revealed that the direct effects of supply chain management (SCM) practices on business performance (BP) is statistically non-significant when controlling the mediator competitive advantage (CA) (b=.1080, t (283) = , p = , which is greater than p value 0.05), means supply chain management (SCM) practices no longer predicts Y or is lessened predicting Y i.e. path c. The Sobel test statistic was reported at z =.1207, with significant p value at p<0.01. Hence, it can be concluded that there is an evidence of complete mediation. And this is depicted in the following diagram. Structural Equation Model The further substantiation could be made if the researchers are interested in using structural equation model to probe the indirect effects are 46

47 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research statistically significant (Shrout and Bolger, 2002). The programs AMOS, R Lavaan, LISREL can perform structural equation models, and the more programs with newer versions are introduced with more compatibility. Here considering the same example, it is not necessary to develop two models with mediator and without mediator, according to Kenny (2018); because the program AMOS is more compatible to give the results of Direct and Indirect Effects. Also, the total effects c could be seen by calculating the value for c + ab. For the example discussed in this paper structural equation model was performed by bootstrapping 2000 samples using biased-confidence interval method. It would be good to use structural equation modeling for causal research as other models like Sobel test, and Process macro uses unstandardized coefficients of regression. Also, it could be noted that there is no standard suggestions for bootstrap sample numbers for obtaining accurate results with standard errors (Nevitt and Hancock, 2001); but if the number of bootstrap samples are high, there is possibility for holding good statistical power (Davidson and Mackinnon, 2000). Usually, the bootstrap works well with n = 300 sample size (Ishikawa and Konishi, 1995), the present study has samples closer to 300. The bootstrap of 2000 samples was considered and used here based on the assumption that non-normality may exist at different conditions. 47

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54 Dr. Meena Madhavan Interpretation (For the example discussed in this paper) The estimates revealed that the effect of supply chain management on business performance is non-significant with a p-value > 0.05 (Refer to table 5.1). The model converged with a chi-square χ(62) value of at p 0.05 (Refer to Figure 1). To analyze the existence of mediation, bootstrapping results were considered. The indirect effect of supply chain management practices on business performance revealed that b =.093, Standard error =.030, confidence interval values of lower bound and upper bound is.045 and.164, which is different from zero and the indirect effects are statistically significant at p 0.01 (Refer to table 5.2). Hence, it proves the evidence of mediation and the indirect effect of supply chain management practices on business performance is statistically significant. In order to assess the type of mediation existing direct effects were considered. The direct effect of supply chain management practices on business performance revealed that b =.041, Standard error =.049, confidence interval values of lower bound and upper bound is and.143, which is zero or negative and the direct effects are statistically non-significant at p 0.05 (Refer to table 5.3). The direct effect of supply chain management practices on firm performance is nonsignificant, and this confirms the evidence of complete mediation. Hence, the relationship between supply chain management practices and business performance advantage is mediated by competitive advantage. The model is fit at all respect according to the threshold levels suggested by Hu and Bentler (1999) for GFI (Figure 5.1), AGFI, RMSEA, RMR, and SRMR. Table 5.1 Estimates Paths Estimate S.E. C.R. P Label CompAdv <--- SCMP *** par_11 BusinPerf <--- CompAdv *** par_12 BusinPerf <--- SCMP par_13 54

55 A Basic Understanding to Mediation Analysis and Statistical Procedures in Management Research The researchers may write in detail about model fit indices. Here it is not given in detail as the objective is to explain the indirect effects. Discussion For performing the mediation tests it is important for the researchers to frame the hypothesis based on the theory and practical applicability, otherwise the researchers may get negative results as no mediation. In this article, an example has been used with dummy dataset to explain the statistical procedures with interpretation. In this article, Firstly, Baron and Kenny s approach was demonstrated through series of regression analysis i.e. simple linear regression. Secondly, hierarchical regression was performed to test the multicollinearity effects, and reported that there is no multicollinearity effects. Because, for analyzing the mediation effect it is 55

56 Dr. Meena Madhavan important to check the multicollinearity issues. Thirdly, indirect effects and the calculation of indirect effects using difference of coefficients and products of coefficients approach were discussed. Also, Sobel test and other versions of the related test were performed to analyze the mediation and its statistical significance. Fourthly, Bootstrapping method was utilized. Bootstrapping is a popular method in testing the mediation (Shrout and Bolger, 2002). Bootstrap method was initially published by Efron (1979), later the statisticians in the field developed the extensions like improved estimates of the variance, Bayesian approaches, Bias-Corrected, and Biascorrected and accelerated bootstrap, etc. It is resampling the sample data to control the stability of the results. It was performed using process macro by professor (Hayes, 2013). The bootstrapping method helps to estimate the confidence interval for indirect effect. That s the reason bootstrapping method is strongly recommended by Hayes (2013) for mediation analysis. Based on the example dummy dataset used in this article, the results of direct, indirect effects and Sobel test were reported appropriately. Fifthly, Structural Equation Modeling was performed to analyze the Direct, Indirect and Total Effects using Bootstrapping method and the results of the sample data were reported appropriately. Hence, five approaches (Baron and Kenny, Difference of coefficients, Products of coefficients, Sobel Test, and Bootstrapping) were used to estimate the indirect or mediation effects of the variables used in this example. Although, each approach has its own criticisms, this article with step-by-step procedures helps the beginners to develop the understanding towards mediation effects. Kenny, Kashy, and Bolger (1998) restated that four step procedures should be undertaken for testing the mediation. Researcher s like Collins, Graham and Flaherty (1998) questioned the Baron and Kenny s first step i.e. testing the relationship between X and Y when the researchers are supposed to test the mediation. Also, it is possible to find the total effect by calculating c + ab. Still, logically the question raised by Collins, Graham and Flaherty (1998) seems to be fine; however the researchers might be interested in framing and testing the hypothesis for X to Y without the mediator to probe the theory. Shrout and Bolger (2002) stated that first step of Baron and Kenny s approach 56